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Documents  Kotschick, Dieter | enregistrements trouvés : 6

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Research talks;Algebraic and Complex Geometry

This is joint with Adrian Langer. Let $X$ be a smooth complex projective variety. We show that every rigid integral irreducible representation $ \pi_1(X,x) \to SL(3,\mathbb{C})$ is of geometric origin, i.e. it comes from a family of smooth projective varieties. The underlying theorem is a classification of VHS of type $(1,1,1)$ using some ideas from birational geometry.

14F35 ; 14D07 ; 58E20 ; 22E40

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Research talks;Algebraic and Complex Geometry;Topology

In ’84, Hirzebruch constructed a very explicit noncompact ball quotient manifold in the process of constructing smooth projective surfaces with Chern slope arbitrarily close to 3. I will discuss how this and some closely related ball quotients are useful in answering a variety of other questions. Some of this is joint with Luca Di Cerbo.

14M27 ; 32Q45 ; 57M50

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Research talks;Algebraic and Complex Geometry

Given an algebraic variety defined by a set of equations, an upper bound for its dimension at one point is given by the dimension of the Zariski tangent space. The infinitesimal deformations of a variety $X$ play a somehow similar role, they yield the Zariski tangent space at the local moduli space, when this exists, hence one gets an efficient way to estimate the dimension of a moduli space.
It may happen that this moduli space consists of a point, or even a reduced point if there are no infinitesimal deformations. In this case one says that $X$ is rigid, respectively inifinitesimally rigid.
A basic example is projective space, which is the only example in dimension 1. In the case of surfaces, infinitesimally rigid surfaces are either the Del Pezzo surfaces of degree $\ge$ 5, or are some minimal surfaces of general type.
As of now, the known surfaces of the second type are all projective classifying spaces (their universal cover is contractible), and have universal cover which is either the ball or the bidisk (these are the noncompact duals of $P^2$ and $P^1 \times P^1$ ), or are the examples of Mostow and Siu, or the Kodaira fibrations of Catanese-Rollenske.
Motivated by recent examples constructed with Dettweiller of interesting VHS over curves, which we shall call BCD surfaces, together with ingrid Bauer, we showed the rigidity of a class of surfaces which includes the Hirzebruch-Kummer coverings of the plane branched over a complete quadrangle.
I shall also explain some results concerning fibred surfaces, e.g. a criterion for being a $K(\pi,1)$-space; I will finish mentioning other examples and several interesting open questions.
Given an algebraic variety defined by a set of equations, an upper bound for its dimension at one point is given by the dimension of the Zariski tangent space. The infinitesimal deformations of a variety $X$ play a somehow similar role, they yield the Zariski tangent space at the local moduli space, when this exists, hence one gets an efficient way to estimate the dimension of a moduli space.
It may happen that this moduli space consists of a ...

14J29 ; 14J80 ; 14P25 ; 32G05

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Research talks;Algebraic and Complex Geometry

We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic group are of general type.

14D07 ; 14F05 ; 14K10 ; 11G18

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Research talks;Algebraic and Complex Geometry

In this talk we consider the Milnor fiber F associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the roots of the Alexander polynomial and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms. The case of line arrangements, where there are many open questions, will illustrate the complexity of the problem. These results are based on joint work with Morihiko Saito, and with Gabriel Sticlaru. In this talk we consider the Milnor fiber F associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the roots of the Alexander polynomial and, in many cases, explicit bases for the ...

32S55 ; 32S35 ; 32S22

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- 524 p.
ISBN 978-4-931469-48-8

Advanced studies in pure mathematics , 0052

Localisation : Collection 1er étage

géométrie différentielle # groupe de difféomorphisme

57-06 ; 19D06 ; 19D10 ; 37C85 ; 37D40 ; 37E10 ; 37E25 ; 37E30 ; 53D05 ; 53D10 ; 53D12 ; 53D35 ; 53D40 ; 53D45 ; 55P35 ; 55P47 ; 55P62 ; 55R35 ; 55R40 ; 57R17 ; 57R19 ; 57R20 ; 57R30 ; 57R32 ; 57R50 ; 57S05 ; 57S25 ; 57S30 ; 58H10

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